One of the tasks that we
assign ourselves in this paper is to define a general notion of
“localization” of an MSN and to prove existence results in some
cases.
Since a definition of “localization” will involve a universal
property, it is critical to determine which category is appropriate
for our purposes.
As this offers unexpected surprises, we describe the several steps in
some detail.
The objects of our first category \(\mathsf {MSN}\)  are the saturated MSN
\((X,{A},{N})\) , by what we mean that for every \(N,N^{\prime } \subseteq X\) , if
\(N \subseteq N^{\prime }\)  and \(N^{\prime } \in {N}\) , then \(N \in {N}\) .
This is in analogy with the notion of a complete measure space.
In order to define the morphisms between two objects \((X,{A},{N})\) 
and \((Y,{B},{M})\) , we say that a map \(f \colon X \rightarrow Y\)  is
\([({A},{N}),({B},{M})]\) -measurable if \(f^{-1}(B) \in {A}\) 
for every \(B \in {B}\)  and \(f^{-1}(M) \in {N}\)  for every \(M \in {M}\) .
For instance, if \(X\)  is a Polish space and \(\mu \)  is a diffuse
probability measure on \(X\) , there exists [1]} a
Borel isomorphism \(f \colon X \rightarrow [0,1]\)  such that \(f_\# \mu = {L}^1\) ,
where \({L}^1\)  is the Lebesgue measure, thus \(f\)  is
\([({B}(X),{N}_\mu ),({B}([0,1]),{N}_{{L}^1})]\) -measurable.
We define an equivalence relation for such measurable maps \(f,f^{\prime } \colon X\rightarrow Y\)  by saying that \(f \sim f^{\prime }\)  if and only if \(\lbrace  f \ne f^{\prime } \rbrace  \in {N}\) .
The morphisms in the category \(\mathsf {MSN}\)  between the objects
\((X,{A},{N})\)  and \((Y,{B},{M})\)  are the equivalence classes of
\([({A},{N}),({B},{M})]\) -measurable maps.
At this stage, we need to suppose that \((X, {A}, {N})\)  is
saturated for the relation of equality almost everywhere to be
transitive REF . With this assumption, the composition of
measurable maps is also compatible with \(\sim \) , see REF .
